Spontaneous Solitons in the Thermal Equilibrium of a Quasi-1D Bose Gas

Tomasz Karpiuk,1,2 Piotr Deuar,3 Przemysław Bienias,4 Emilia Witkowska,3 Krzysztof Pawłowski,4,5 Mariusz Gajda,3

Kazimierz Rzążewski,4,5 and Mirosław Brewczyk1

1Wydział Fizyki, Uniwersytet w Białymstoku, Lipowa 41, 15-424 Białystok, Poland2Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore

3Institute of Physics PAN, Aleja Lotników 32/46, 02-668 Warsaw, Poland4Center for Theoretical Physics PAN, Aleja Lotników 32/46, 02-668 Warsaw, Poland

5Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany(Received 13 June 2012; revised manuscript received 19 September 2012; published 13 November 2012)

We show that solitons occur generically in the thermal equilibrium state of a weakly interacting

elongated Bose gas, without the need for external forcing or perturbations. This reveals a major new

quality to the experimentally widespread quasicondensate state, usually thought of as primarily phase-

fluctuating. Thermal solitons are seen in uniform 1D, trapped 1D, and elongated 3D gases, appearing as

shallow solitons at low quasicondensate temperatures, becoming widespread and deep as temperature

rises. This behavior can be understood via thermal occupation of the type II excitations in the Lieb-Liniger

model of a uniform 1D gas. Furthermore, we find that the quasicondensate phase includes very

appreciable density fluctuations while leaving phase fluctuations largely unaltered from the standard

picture derived from a density-fluctuation-free treatment.

DOI: 10.1103/PhysRevLett.109.205302 PACS numbers: 67.85.�d, 03.75.Hh, 03.75.Lm, 05.45.Yv

Solitons, or nondestructible local disturbances, are im-portant features of many one-dimensional (1D) nonlinearwave phenomena. In ultracold gases, they have long beensought and were first observed to be generated by phaseimprinting [1,2]. More recently, their spontaneous forma-tion in 1D gases was predicted as a result of the Kibble-Zurek mechanism [3,4], rapid evaporative cooling [5], anddynamical processes after a quantum quench [6]. Here weshow that they actually occur generically in the thermalequilibrium state of a weakly interacting elongated Bosegas, without the need for external forcing or perturbations.This reveals a major new quality to the experimentallywidespread quasicondensate state. It can be understood viathermal occupation of the famous and somewhat elusivetype II excitations in the Lieb-Liniger model of a uniform1D gas [7].

A mathematically distinct class of soliton equations arethe completely integrable systems. Among them, theGross-Pitaevskii [8,9] equation describes weakly interact-ing bosons in a 1D geometry in the mean field approxima-tion. The corresponding multiatom Lieb-Liniger modelof N bosons on the circumference of a circle interactingby contact forces [10] has elementary excitations of twokinds: those of a Bogoliubov type and an additional type IIbranch [7]. These additional excitations have been associ-ated with solitons of the mean field [11–14]. Although thetrapping potential removes integrability, from the earlydays experimenters have searched for gray solitons. SeeRef. [15] for a review.

Typically, by irradiating one part of the condensate, oneengineers a phase difference with the remainder, and a darksoliton forms at the interface between the phase domains

[1,2]. Other proposed schemes involve taking the systemaway from equilibrium [3–6]. Our results show that sol-itons are in fact present spontaneously even in equilibrium.However, engineered solitons have been easier to identifywith the standard destructive imaging measurements be-cause their position does not vary from shot to shot.Here we generate a classical field ensemble that de-

scribes the weakly interacting Bose gas at thermal equilib-rium [16] and then show that gray solitons indeed arealready there. This is demonstrated by tracing the timeevolution of a single copy of the system. We find spectralproperties consistent with Lieb type II excitations.Moreover, we will also show that the presence of graysolitons in thermal equilibrium remains valid for veryelongated traps that are no longer strictly 1D.First, let us consider the simplest situation: a uniform

1D weakly repulsively interacting gas in free space withperiodic boundary conditions, as per Lieb and Liniger [10].To obtain the equilibrium state, we apply the classical fieldapproximation (CFA) [17,18]. Within this approach, the

usual bosonic field operator �̂ðzÞ which annihilates anatom at point z is replaced by an ensemble of complexwave functions �ðzÞ obtained by using a Monte Carlosampling algorithm [16] (see Supplemental Material [19]for details ).In Fig. 1, we show the time evolution of the density and

phase for single realizations in equilibrium at severaltemperatures. The starting conditions are randomly chosenfrom the collection of wave functions for the canonicalensemble obtained in the CFA. Subsequent time evolutionof these chosen realizations in the CFA is via the Gross-Pitaevskii equation of motion.

PRL 109, 205302 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 NOVEMBER 2012

0031-9007=12=109(20)=205302(5) 205302-1 � 2012 American Physical Society

http://dx.doi.org/10.1103/PhysRevLett.109.205302

The only relevant dimensionless parameter appearingin the Lieb-Liniger model of a 1D interacting Bose gas is� ¼ mg=@2�, where g characterizes the atom-atom inter-action and � ¼ N=L is the linear density of the system.In what follows, we will use L, mL2=@, @2=mL2, and@2=mL2kB as the units of length, time, energy, and tem-

perature, respectively.First, the top panels show the system at the relatively low

T ¼ 104 ¼ �=2. There are numerous density disturbancesin the form of dips and peaks, traveling in both directions,near the speed of sound (which is depicted by the super-imposed red line). There exist both defects that travel fasterand those that travel slower than sound. The deepest dipsare slower—the slope of their trajectories on the plot ishigher than that of the superimposed red line. At lowtemperature only fast-moving shallow defects are present.Some of them correspond to packets of Bogoliubov pho-nons, and others are solitons. This can be checked byinspection of the phase jump across the defect (for a solitonits sign is related to its direction of movement) and byfitting the local density dip to a soliton solution [20].However, shallow fast solitons can be hard to tell fromphonons.

For a higher temperature T ¼ 7� 104 (middle panels),long-lived deep dips traveling far slower than the speed ofsound appear. These become prolific as the temperature

increases to T ¼ 10� 104 (lower panel). An analysis ofthe parameters of slower-than-sound defects confirmstheir interpretation as dark solitons. Indeed, one sees theanticorrelation between the depths and speeds of the dipsthat is expected for dark solitons. Near maximal depth, thedips approach being stationary. One also sees numeroussoliton collisions, and associated phase shifts, with thenumber of solitons usually conserved.The match between dark solitons and Lieb type II ex-

citations is seen by comparing fitted parameters describingthe local density dips to the gray soliton solutions. To dothis, we fit a density profile (characterized by soliton depth,local density, and soliton position) to each dip. If the rmsparameter errors are smaller than 10% of the local density(for depth or density) or the healing length (for position),we accept the dip as a soliton. We then calculate solitonenergy � and momentum p from the parameters [11],collecting data from many time snapshots. The resultingenergy-momentum relation of the accepted dips is com-pared with the Lieb type II excitation spectrum in Fig. 2 forthree temperatures. At the lowest temperature, the spectralmatch is good and includes the negative curvature of thetype II branch. For higher temperatures, the fitted solitonenergy lies somewhat below the Lieb II line, retaining aqualitative match. Interpretation of the shift is difficult, asinteractions between solitons are frequent at this tempera-ture. Notably, one sees a filling of the high p spectrum withtemperature.This allows us to explain the temperature dependence of

the appearance of deep solitons. These have the maximumLieb II energy and momentum [13]. For the parameters ofFig. 1, that is about 105 in units of temperature (seeSupplemental Material [19]). Thus we would expect thedeepest solitons to appear when temperatures of this orderare reached. This is what is seen. We found similar agree-ment for � ¼ 0:002.It is appealing to think of the system as of a gas of

quasiparticles of two kinds—where two bosonic excitation

FIG. 1 (color online). Density (left) and phase (right) as afunction of time of a 1D Bose gas at equilibrium for a singlerealization. Here � ¼ 0:02 (N ¼ 103), and the temperatures areT ¼ 104 ¼ �=2 (top), 7� 104 (middle), and 105 (bottom) inunits of @2=mL2kB. The superimposed red line in the upper leftpanel corresponds to travel at the speed of sound.

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.05

0.10

0.15

0.20

p

2

FIG. 2 (color online). Comparison of the spectra of Lieb typeII excitations (solid black line) and dips in the thermal statefor three different temperatures: T ¼ 104 (medium red dots),4� 104 (light green dots), and 105 (dark blue dots). Here � ¼0:02 (N ¼ 103).

PRL 109, 205302 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 NOVEMBER 2012

205302-2

families coexist like in the Lieb-Liniger model (seeSupplemental Material [19]). These two kinds—phononsand solitons—interact with each other in an inelastic way,so that solitons can be born from Bogoliubov excitationsand vice versa. Such interactions could allow the gas ofquasiparticles to reach the equilibrium state.

Now a question that begs to be asked is whether solitonswill also be present in the less idealized case of a trappedgas in the quasicondensate regime—a situation ubiquitousin contemporary experiments. The CFA can be applied ina similar way as above (see Supplemental Material [19]).First, consider a harmonically trapped 1D gas. Figure 3(left) shows its density evolution for two different tem-peratures. The lower temperature case (left upper panel)has T chosen such that the phase coherence length l� is

approximately equal to the size of the cloud W; i.e., itcorresponds to the temperature T�, which separates the

quasicondensate and true condensate phases, as first de-scribed by Petrov, Shlyapnikov, and Walraven [21]. Fortemperatures lower than this, only shallow fast solitons areseen in the bulk of the cloud. At much higher temperature(lower left panel in Fig. 3), a mass of deeper long-livedsolitons emerges. They remain within the main part of theatomic cloud, and their number and depth increase with

temperature. One can easily check that for the lower tem-peratures the solitons oscillate in the trap with a frequency

which within a few percent agrees with !=ffiffiffi2

pfrom a

simple model [22]. At high temperatures, intersoliton col-lisions come to dominate the oscillating behavior, makingthat model inappropriate.The center of the cloud can be considered as a uniform

gas under a local density approximation as confirmedin recent experiments [23,24]. This is described by thedimensionless interaction strength (� ¼ g=hni) and tem-perature (� ¼ 2kBTm=@2hni2). In Figs. 3 and 4, the T ¼ 15gas lies deep in the quasicondensate regime (� ¼ 0:0033,� ¼ 0:0034), while the center of the T ¼ 260 cloud is adecoherent quantum gas [25] (� ¼ 0:0059, � ¼ 0:19).Here, Fig. 4 shows that significant local density fluctua-tions �n=hni � 10–40% are present in the quasicondensatephase � � ffiffiffiffi�p . This is to be contrasted with the commonview that the only notable fluctuations in the quasiconden-sate are those of the phase, a situation we see only at thevery lowest temperatures. (Note also the long-wavelengthdensity fluctuations in Ref. [24]).Importantly, the CFA fluctuations agree with predictions

obtained from the exact local density correlation function[25] in the Yang-Yang description [26] of the interactinguniform gas. Note that we have taken account of the factthat the CFA does not include zero temperature fluctua-tions, so that the �n in Fig. 4 shows only the thermal

FIG. 3 (color online). Solitons in trapped clouds. The leftpanels show the time evolution of a trapped 1D condensatewith parameters: N ¼ 1000, 1D interaction strength g ¼ 0:31in units of @!zð@=m!zÞ1=2, and trap frequency !z ¼2�� 10 Hz. The temperatures are kBT ¼ 15@!z (upper frame)and kBT ¼ 260@!z (lower frame). The right panel shows theevolution of the central density in a three-dimensional elongatedtrapped cloud (horizontal direction). Here, the trap frequenciesare !z ¼ 2�� 10 Hz and !? ¼ 2�� 1000 Hz, with againN ¼ 1000, and the (3D) interaction strength g3D ¼ 0:213 inunits of @!zð@=m!zÞ3=2. The temperature is 80@!z=kB.

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250T

T=T

φ

τ =γ1

/2therm

al δn /

lφ / W

Yang&Y

angCFA

BE

C

Quasicondensate

Decoherentquantumregime

FIG. 4 (color online). Phase and thermal density fluctuations inthe center of the trapped 1D gas as a function of temperature.Parameters are as in Fig. 3 except for varying T (in units of@!z=kB). Purple circles show �n=hni in the center of the trap,orange squares the ratio of the phase coherence length l� to the

cloud width W. The latter is taken to be the distance betweenpoints at which the density has fallen to 10% of the central value.Vertical gray bars indicate temperatures T� and an estimate of

the location of the quasicondensate–decoherent-quantum-gascrossover at � ¼ ffiffiffiffi�p [25]. Yang-Yang predictions (black solidline) are for the average density in the center (jzj � 2) of thecloud. Some statistical uncertainty is visible.

PRL 109, 205302 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 NOVEMBER 2012

205302-3

contribution. The equivalent contribution in the Yang-Yang

solution is ð�nÞthermal=hni ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigð2Þð0Þ � gð2ÞT¼0ð0Þ

q, which is

shown as a black line. The level of agreement implies thatthe solitons we have seen here in Fig. 3, that contribute alarge proportion of the density fluctuations, must also be anessential feature of the Yang-Yang description.

We have also calculated the coherence length through

matching the phase correlation function gð1Þðz; z0Þ ¼�1ðz; z0Þ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1ðz; zÞ�1ðz0; z0Þ

pto an exponential decay

exp½�jz� z0j=l��. This is shown in Fig. 4 in orange andis still in agreement with the canonical treatment thatignores density fluctuations [21]. The one-particle densitymatrix �1ðz; z0Þ is obtained by an averaging of h��ðzÞ�ðz0Þiover the initial canonical ensemble. For the low temperature

case (T ¼ 15), l� ¼ 13:7a0 ¼ 13:7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@=!zm

p, while the

cloud width W is 14:7a0. To compare, the standard quasi-condensate expressions in Ref. [21] that consider onlyphase fluctuations give an estimate of the temperature at

which l� ¼ W. It is Tphaseonly� ¼ 3Nð@!zÞ2

8�kB¼ 12:5@!z=kB for

these parameters, which is a good agreement with theCFA calculation. Our agreement on T� with the standard

description is so good because it is a manifestation ofthe fact that the correlation length obtained from a pure

phase dependence of the classical field, i.e., gð�Þðz; z0Þ ¼expfi½�ðzÞ ��ðz0Þ�g like in Ref. [21]—where �ðzÞ is thephase at point z—does not differ significantly from what is

obtained via the complete correlation function gð1Þðz; z0Þ[27]. We conclude then that the 10–40% density fluctua-tions and solitons that we see in the quasicondensate are notinconsistent with the past calculations of phase fluctuations.Rather, they are an inherent feature that had remainedunnoticed until this time due to the prevalent focus on phasecoherence.

These results raise an interesting issue regarding theKibble-Zurek mechanism of defect formation in 1D gases.In a recent paper [3], it was conjectured that dark solitonsshould be created by the Kibble-Zurek mechanism duringrapid cooling of the gas [28,29]. Simulations [5] of thecooling of a 1D Bose gas in a harmonic trap indeed revealedthe presence of solitons. However, we know now that somesolitons are present even in thermal equilibrium. Only in thecase when the number of solitons at the end of the coolingprocess exceeds that found in its corresponding thermalequilibrium could simulations of that kind confirm theKibble-Zurek mechanism at work.

Finally, there is a question of whether these solitons canstill survive when the strict 1D trapped system crosses overto one in a very elongated cigar-shaped trap. This is madeplausible by past workwhich showed that quasicondensate-BEC-like transitions occur for cigar-shaped systems anal-ogously with the true 1D gas [30–32]. The right panel ofFig. 3 shows the evolution of a thermal state in a fully three-dimensional calculation. Solitons are still clearly visible.

This indicates that they may be present, and indeed evenwidespread, in many existing experiments.The observation of thermal solitons with existing equip-

ment has been nontrivial because of two factors: Thesoliton width in situ was typically significantly narrowerthan the detector resolution, and an identification of sol-itons by eyeballing is only straightforward if one hasaccess to observations of the in situ dynamics, not destruc-tive snapshots. Looking at the edges of the gas offers anindirect way of detecting the solitons [33]. As they bouncebetween the edges of the gas, they approximately retaintheir absolute depth while also increasing in width andreaching close to 100% relative depth at the turning point,making them accessible with realistic resolutions. We donot see correspondingly deep phonon disturbances in ourimages. Therefore [33], we have counted the density ofdips with over 90% relative depth in the wings of the 1Dcloud. The growth of the density of these solitons withtemperature is shown in Fig. 5.Further progress could be made by recent experimental

advances such as high-resolution detection based on scat-tering of an electron beam [34], analysis of momentumspectra [6], phase-jump statistics [35] (phase domains arelarger than soliton dips), or with long [1,2,36] or anti-trapped expansions [37] (see Supplemental Material [19]for more detail).In summary, we have found that solitons are a natural

and spontaneous feature of the 1D and elongated weaklyinteracting Bose gases. While solitons have been studied insuch systems before by imprinting, the above results showthat they are in fact common. The good agreement with

FIG. 5 (color online). Number of solitons per micrometer onthe edges of the cloud, per snapshot, versus temperature. Thereare 104 87Rb atoms in a 1D trap with !z ¼ 2�� 1:9 Hz. The1D coupling assumes a Gaussian transverse profile correspond-ing to !? ¼ 2�� 128 Hz. Counting was over 7 �m intervalson either side of the cloud centered around the position where theaverage density is 10 �m�1 (12% of peak). Dips whose centraldensity falls to below 10% of the mean density at their trapposition were counted.

PRL 109, 205302 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 NOVEMBER 2012

205302-4

exact density fluctuations in Fig. 4 shows that quantitativestudy of these phenomena is possible with the CFA andpossibly related methods. The 1D Bose gas at thermalequilibrium is a system where two very different kinds ofbosonic excitations are present simultaneously. The systemactually contains much more than the standard picture,especially in the quasicondensate regime. Here, in additionto the well known phase fluctuations, appreciable densityfluctuations are found, including deep solitons. Thesespontaneous solitons in the thermal state are analogous topairs of vortices present in a 2D gas near the Berezinskii-Kosterlitz-Thouless transition [38].

We are grateful to Matthew Davis, Tilman Pfau, andTomasz Świsłocki for helpful discussions. T. K. acknowl-edges support by the National Science Center GrantNo. 2011/01/B/ST2/05125, P. D. by the Polish Governmentresearch Grant No. N N202 128539, and P. B., K. P., andK. R. by Polish Government research Grant No. N N202174239 both for the years 2010–2012. M.G., E.W., andM.B. acknowledge support from the EU NAMEQUAMproject, and P. B., K. P., and K. R. acknowledge financialsupport by contract research ‘‘Internationale Spitzen-forschung II-2’’ of the Baden-Württemberg Stiftung,‘‘Decoherence in long range interacting quantum systemsand devices.’’ The CQT is a Research Centre of Excellencefunded by the Ministry of Education and the NationalResearch Foundation of Singapore.

[1] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, and K.Sengstock, Phys. Rev. Lett. 83, 5198 (1999).

[2] J. Denschlag et al., Science 287, 97 (2000).[3] W.H. Zurek, Phys. Rev. Lett. 102, 105702 (2009).[4] B. Damski and W.H. Zurek, Phys. Rev. Lett. 104, 160404

(2010).[5] E. Witkowska, P. Deuar, M. Gajda, and K. Rzążewski,

Phys. Rev. Lett. 106, 135301 (2011).[6] M. Schmidt, S. Erne, B. Nowak, D. Sexty, and T.

Gasenzer, New J. Phys. 14, 075005 (2012).[7] E. H. Lieb, Phys. Rev. 130, 1616 (1963).[8] L. P. Pitaevskii, Zh. Eksp. Teor. Fiz. 40, 646 (1961).[9] E. P. Gross, Nuovo Cimento 20, 454 (1961).[10] E. H. Lieb and W. Liniger, Phys. Rev. 130, 1605

(1963).[11] P. P. Kulish, S. V. Manakov, and L.D. Faddeev, Theor.

Math. Phys. 28, 615 (1976).[12] E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Qi,

Phys. Rev. Lett. 85, 1146 (2000).

[13] A. D. Jackson and G.M. Kavoulakis, Phys. Rev. Lett. 89,070403 (2002).

[14] R. Kanamoto, L. D. Carr, and M. Ueda, Phys. Rev. A 81,023625 (2010).

[15] D. J. Frantzeskakis, J. Phys. A 43, 213001 (2010).[16] E. Witkowska, M. Gajda, and K. Rzążewski, Opt.

Commun. 283, 671 (2010).[17] M. Brewczyk, M. Gajda, and K. Rzążewski, J. Phys. B 40,

R1 (2007).[18] P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and

C.W. Gardiner, Adv. Phys. 57, 363 (2008).[19] See Supplemental Material at http://link.aps.org/

supplemental/10.1103/PhysRevLett.109.205302 for addi-

tional information.[20] V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 64,

1627 (1973) Sov. Phys. JETP 37, 823 (1973).[21] D. S. Petrov, G.V. Shlyapnikov, and J. T.M. Walraven,

Phys. Rev. Lett. 85, 3745 (2000).[22] T. Busch and J. R. Anglin, Phys. Rev. Lett. 84, 2298

(2000).[23] A. H. van Amerongen, J. van Es, P. Wicke, K.

Kheruntsyan, and N. van Druten, Phys. Rev. Lett. 100,090402 (2008).

[24] T. Jacqmin, J. Armijo, T. Berrada, K. Kheruntsyan, and

I. Bouchoule, Phys. Rev. Lett. 106, 230405 (2011).[25] K. V. Kheruntsyan, D.M. Gangardt, P. D. Drummond, and

G.V. Shlyapnikov, Phys. Rev. Lett. 91, 040403 (2003).[26] C. N. Yang and C. P. Yang, J. Math. Phys. (N.Y.) 10, 1115

(1969).[27] P. Bienias,K. Pawłowski, M. Gajda, and K. Rzążewski,

arXiv:1203.1811.[28] T.W.B. Kibble, J. Phys. A 9, 1387 (1976).[29] W.H. Zurek, Nature (London) 317, 505 (1985).[30] D. S. Petrov, G.V. Shlyapnikov, and J. T.M. Walraven,

Phys. Rev. Lett. 87, 050404 (2001).[31] D. Kadio, M. Gajda, and K. Rzążewski, Phys. Rev. A 72,

013607 (2005).[32] S. Dettmer et al., Phys. Rev. Lett. 87, 160406 (2001).[33] Following a suggestion of M. Oberthaler.[34] V. Guarrera, P. Würtz, A. Ewerbeck, A. Vogler, G.

Barontini, and H. Ott, Phys. Rev. Lett. 107, 160403(2011).

[35] T. Betz et al., Phys. Rev. Lett. 106, 020407 (2011).[36] C. Becker, S. Stellmer, P. Soltan-Panahi, S. Dörscher, M.

Baumert, E.-M. Richter, J. Kronjäger, K. Bongs, and K.

Sengstock, Nat. Phys. 4, 496 (2008).[37] J. J. Chang, P. Engels, and M.A. Hoefer, Phys. Rev. Lett.

101, 170404 (2008).[38] V. L. Berezinskii, Zh. Eksp. Teor. Fiz. 61, 1144 (1971);

J.M. Kosterlitz and D. J. Thouless, J. Phys. C 5, L124(1972).

PRL 109, 205302 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 NOVEMBER 2012

205302-5

http://dx.doi.org/10.1103/PhysRevLett.83.5198http://dx.doi.org/10.1126/science.287.5450.97http://dx.doi.org/10.1103/PhysRevLett.102.105702http://dx.doi.org/10.1103/PhysRevLett.104.160404http://dx.doi.org/10.1103/PhysRevLett.104.160404http://dx.doi.org/10.1103/PhysRevLett.106.135301http://dx.doi.org/10.1088/1367-2630/14/7/075005http://dx.doi.org/10.1103/PhysRev.130.1616http://dx.doi.org/10.1007/BF02731494http://dx.doi.org/10.1103/PhysRev.130.1605http://dx.doi.org/10.1103/PhysRev.130.1605http://dx.doi.org/10.1007/BF01028912http://dx.doi.org/10.1007/BF01028912http://dx.doi.org/10.1103/PhysRevLett.85.1146http://dx.doi.org/10.1103/PhysRevLett.89.070403http://dx.doi.org/10.1103/PhysRevLett.89.070403http://dx.doi.org/10.1103/PhysRevA.81.023625http://dx.doi.org/10.1103/PhysRevA.81.023625http://dx.doi.org/10.1088/1751-8113/43/21/213001http://dx.doi.org/10.1016/j.optcom.2009.10.080http://dx.doi.org/10.1016/j.optcom.2009.10.080http://dx.doi.org/10.1088/0953-4075/40/2/R01http://dx.doi.org/10.1088/0953-4075/40/2/R01http://dx.doi.org/10.1080/00018730802564254http://link.aps.org/supplemental/10.1103/PhysRevLett.109.205302http://link.aps.org/supplemental/10.1103/PhysRevLett.109.205302http://dx.doi.org/10.1103/PhysRevLett.85.3745http://dx.doi.org/10.1103/PhysRevLett.84.2298http://dx.doi.org/10.1103/PhysRevLett.84.2298http://dx.doi.org/10.1103/PhysRevLett.100.090402http://dx.doi.org/10.1103/PhysRevLett.100.090402http://dx.doi.org/10.1103/PhysRevLett.106.230405http://dx.doi.org/10.1103/PhysRevLett.91.040403http://dx.doi.org/10.1063/1.1664947http://dx.doi.org/10.1063/1.1664947http://arXiv.org/abs/1203.1811http://dx.doi.org/10.1088/0305-4470/9/8/029http://dx.doi.org/10.1038/317505a0http://dx.doi.org/10.1103/PhysRevLett.87.050404http://dx.doi.org/10.1103/PhysRevA.72.013607http://dx.doi.org/10.1103/PhysRevA.72.013607http://dx.doi.org/10.1103/PhysRevLett.87.160406http://dx.doi.org/10.1103/PhysRevLett.107.160403http://dx.doi.org/10.1103/PhysRevLett.107.160403http://dx.doi.org/10.1103/PhysRevLett.106.020407http://dx.doi.org/10.1038/nphys962http://dx.doi.org/10.1103/PhysRevLett.101.170404http://dx.doi.org/10.1103/PhysRevLett.101.170404http://dx.doi.org/10.1088/0022-3719/5/11/002http://dx.doi.org/10.1088/0022-3719/5/11/002

Supplementary material for:Spontaneous solitons in the thermal equilibrium of a quasi-one-dimensional Bose gas

Tomasz Karpiuk,1,2 Piotr Deuar,3 Przemysław Bienias,4 Emilia Witkowska,3

Krzysztof Pawłowski,4,5 Mariusz Gajda,3 Kazimierz Rzą̇zewski,4,5 and Mirosław Brewczyk11Wydział Fizyki, Uniwersytet w Białymstoku, ul. Lipowa 41, 15-424 Białystok, Poland

2Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore3Institute of Physics PAN, Al. Lotników 32/46, 02-668 Warsaw, Poland

4Center for Theoretical Physics PAN, Al. Lotników 32/46, 02-668 Warsaw, Poland5Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany

The classical field approximation

Within this approach [1, 2] the usual bosonic field opera-tor Ψ̂(z) which annihilates an atom at pointz is replaced bythe complex wave functionΨ(z). Technically speaking, wefirst expand the field operator in the basis of one-particle wavefunctions, appropriate for the problem considered. Then, ex-tending the original Bogoliubov idea [3] to all macroscopi-cally occupied one-particle modes, we replace the operatorscorresponding to these modes byc-numbers. Restricting theexpansion only to these modes, the field operator is turned intoa complex wave function – the classical field. For a plane-wave basis,

Ψ(z) =∑

|k|≤kmax

αk1√Leikz . (1)

The above expression is appropriate for bosons confined in abox of lengthL with periodic boundary conditions. For a har-monic trapping potential, the single-particle trap eigenfunc-tions are more convenient. The summation is extended overall modes up to the momentum cut-off~kmax. The optimalchoice of the cut-off is discussed in Ref. [4]. Fully three di-mensional, elongated trap simulations require recalculation ofthe optimal cut-off condition since the explicit results in[4]are valid only for symmetric D-dimensional traps and, more-over, are asymptotic for large number of atoms. Each classicalfield,Ψ(z), shares many properties with the single-shot mea-surements of the atomic cloud that occur in experiment. Theclassical field satisfies the following equation of motion [1]:

i~∂

∂tΨ(z, t) =

(

− ~2

2m

∂2

∂z2+ g |Ψ(z, t)|2

)

Ψ(z, t) , (2)

where g characterizes the atom–atom interaction and thenonlinear term is projected on the subspace spanned by themacroscopically populated modes.

Thermal states

To obtain the thermal equilibrium state of a 1D Bose gaswithin the CFA, we numerically generate members of the

canonical ensemble of states [5], i.e. states populated accord-ing to the probability distribution given by:

P ({αk}) =1

Ze−EΨ/kBT , (3)

whereZ is the canonical partition function andT is a temper-ature. The energy,EΨ, accumulated in the classical field isgiven by :

EΨ =

∫

L

dzΨ∗(z)H0Ψ(z) +1

2g

∫

L

dz|Ψ(z)|4 . (4)

whereH0 is the single-particle Hamiltonian

H0 = Vtrap(z)−~2

2m

∂2

∂z2(5)

with the trapping potentialVtrap(z). An extra constraint onthe amplitudes{αk} should be fulfilled:

∑

|k|≤kmax

|αk|2 = N , (6)

whereN is the number of atoms. An efficient way to ob-tain states belonging to the canonical ensemble at given tem-peratureT is a Monte Carlo method using the Metropolisalgorithm[6]. Here, a random walk in the phase space of thesystem is performed and all visited states become the mem-bers of the canonical ensemble. These states are used to cal-culate statistical averages of any observable. Details forultra-cold Bose gas systems are given in Ref. [5].

Temperature at which deep solitons appear

Assuming the excitations observed within the CFA are Liebtype II excitations, we are able to estimate the temperatureatwhich deepest solitons appear in the thermal state of a uniformsystem. To obtain the dispersion curve for the type II excita-tions, one is required to solve the inhomogeneous Fredholmintegral equations [7]. This can be done numerically, and forγ = 0.02 the dispersion curve, i.e. energy versus momentumǫ(p), is found to be well approximated byǫ(p) = aρp+ bp2,wherea = 0.125 andb = −0.021. Maximal excitation energyoccurs for the highest momentum on the curvepmax = πN(see [7] for details), and corresponds to the deepest, stationary

2

solitons. It is then given by the formulaǫmax = πN2(a+ πb)in our units, which forN = 1000 is about105. Therefore, forγ = 0.02, the deepest solitons are expected at temperatures ofthis order or higher.

Spectral analysis

Further arguments for the simultaneous presence of bothkinds of excitations (Type I – Bogoliubov phonons, and TypeII – solitons) in a 1D Bose gas at the equilibrium come from aspectral analysis of the classical field. Fig. 1 shows the space-time spectral density|Ψ(ω, k)|2 of a single realization at equi-librium. Like in the three-dimensional case (see Ref. [8]),twocurves crossing at a frequency equal to the chemical potentialµ = 2× 104 are clearly visible. The low-momentum slope ofeach curve is just the speed of sound. The parts of these curvesbelow the chemical potential are necessary to construct thephonon-like section of the excitation spectrum [8, 9] and fadeat momenta at which the dispersion curve changes its charac-ter from linear to parabolic. This part of the spectrum provesthe existence of Bogoliubov phonons in thermal equilibrium.

Figure S 1. Spectral density of a 1D Bose gas at equilibrium forγ = 0.02, T = 104 within the classical field approximation (up-per panel, detail – lower left panel). In addition to the Bogoliubovphonon nature of the main curves, the spectral density exhibits acheckerboard pattern for low momenta. This pattern is related to thepresence of solitons in the system. Bottom right panel: spectral den-sity of two dark solitons propagating in opposite directions. Solitonvelocities are±0.8c, with c the speed of sound. The checkerboardpattern again appears on top of the main structure consisting of twolinesω = ±uk. It is interpreted as a result of interference betweensolitons.

However, in addition to the phonon-like behavior, thespectral density exhibits a phenomenon absent in the three-dimensional system – in the region of low energies and mo-menta a checkerboard pattern appears (see Fig. 1). This is asignature of the presence of solitons. We confirm this inter-pretation by considering the spectral density of two counter-

propagating dark solitons moving with the same velocityu(and depth), shown in the lower right panel of Fig. 1. Thesesolitons are obtained from the Zakharov solution [10], and itssubsequent evolution according to the nonlinear Schrödingerequation. Finally, the dispersion curve of a single dark solitonconsists of only a single lineω = uk. Hence, the checker-board pattern appears as a result of interference between thesolitons, and thus it is a strong signature of the existence ofdark solitons in the system. Indeed, the checkerboard patternbecomes more regular when the number of solitons increases.

The classical field approximation in one dimension

The classical field approach[1] and the very closely relatedPGPE (Projected Gross-Pitaevskii Equation) approach[2]have been benchmarked on numerous occasions — see de-scription in the reviews [1, 2], or [11]. To demonstrate itsapplicability to the density fluctuations of 1D gases as consid-ered in this article, we have compared CFA results to a recentexperiment that measured density fluctuations[12].

Using the actual parameters of [12] and the prescription forthe cut-off parameter used in this article, we obtained the verygood agreement illustrated in Fig S2.

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45

δ2 n

Figure S 2. Local atom-number variances,δ2n, as a function ofthe mean local atom density in a weakly interacting quasi 1D gasconfined by a harmonic potential. Red circles are taken from Fig.1c. in [12]. Blue crosses are the results of our 1D simulationwith parameters from the experiment [12]:g1D = 2~ω⊥a =2.06~ωz(~/mωz)

1/2 is the coupling constant,a = 5.7nm is the3D s-wave scattering length, andω⊥ = 2π× 3.9 kHz,ωz = 2π× 4Hz are the frequencies of the transverse and longitudinal harmonicconfining potentials, respectively.T = 0.09~ω⊥/kB = 88~ωz/kB .

Possible detection schemes

Let us now turn to the question of experimental observationof solitons in the thermal state. Unfortunately, they are notstraightforward to directly detect because of two factors:(1)

3

The soliton width in-situ, being a fraction of aµm, is signif-icantly narrower than the usual detector resolution (typicallyseveralµm). However, a recently developed detection schemebased on scattering of an electron beam could overcome thisdifficulty [13]. Also, (2) an identification of solitons by “eye-balling” is only straightforward if one has access to observa-tions of the in-situ dynamics – a single-time density slice typi-cally shows many density dips, and it is not easy to distinguishthe fast “phonon-like” ones from bona-fide solitons.

103

104

-1 -0.5 0 0.5 1

resolution: 4.4 ξ

103

104

105

-1 -0.5 0 0.5 1phase jump between pixels / π

resolution: 1.1 ξ

103

104

105

106

107

-1 -0.5 0 0.5 1

freq

uenc

y

resolution: 0.28 ξ

101

102

103

104

-1 -0.5 0 0.5 1

resolution: 8.8 ξ

100

101

102

103

104

105

-0.4 -0.2 0 0.2 0.4

resolution: 1.1 ξ

100

101

102

103

104

105

106

-0.1 -0.05 0 0.05 0.1

freq

uenc

y

resolution: 0.28 ξ

Figure S 3. Experimental signatures of thermal solitons in the uni-form gas — phase-jump frequency (log scale) atγ = 0.02 for lowtemperature (no deep solitons – blue,T = 104) and high tempera-ture (many deep solitons – red,T = 10 × 104). Columns show theeffect of different imaging resolutions in units of healinglengthξ (= 1/N

√γ in our units), which is the half-width of a deep soliton.

Resolution worsens from left (belowξ) to right (unable to resolvesoliton dips). Note the strong resistance of the differencein shapeof the distribution (parabolic / extended) even into regimes whereindividual solitons cannot be resolved.

However, several approaches hold promise of overcomingone or both of the above issues. One is a direct observation ofdeep solitons in an expanding cloud. After about10ms of ex-pansion of a typical cloud, the soliton size can exceed detectorresolution. The method was successfully used in engineeredsoliton experiments [14–16].

Another promising approach is to look for the expectedlarge phase jump between phase domains that occurs at thesoliton. If relatively recognizable phase domains are presentbetween solitons, then the phase jump of close to±π thatoccurs at deep soliton defects should be detectable with animaging resolution that is sufficient only to resolve the phasedomains [17]. These are much wider than the solitons if thesoliton density is not extreme.

It turns out that even with very noisy domains, a qualitativedifference can be observed between clouds with and without

solitons. Fig. S3 shows the phase-jump histogram (for phasejumps between neighboring pixel pairs) atγ = 0.02 for bothlow temperature and high temperature. The low-temperaturesystem has a Gaussian distribution of phase jumps (parabolicon the plot, which has a log scale), which broadens but doesnot change shape as the resolution is worsened. This reflectsthe addition of more and more random small phase fluctu-ations that arise primarily from Bogoliubov excitations. Instark contrast, when deep solitons are present, phase jumpsofπ appear quite frequently, and – most significantly – the distri-bution flattens out for large jumps. This qualitative difference(flattening-out / Gaussian) survives even to resolutions that areincapable of resolving the actual density dip of the solitons.

The anti-trapping technique may also be quite promising.This technique has been used to image short wavelengthshock waves in elongated gases [18], which have some phe-nomenological features in common with solitons. An invertedparabolic potential is rapidly applied to the gas along the longdirection to prevent the evolution of in-situ density to momen-tum density in that direction, but to instead magnify the den-sity profile.

Finally, a recent work has presented signatures of soli-tons in the form of characteristic features of the momentumdistribution[19]. It analyzed solitons formed after a quantumquench, but is also appropriate for the randomly placed ther-mal solitons discussed here.

[1] M. Brewczyk, M. Gajda, and K. Rzążewski, J. Phys. B40, R1(2007).

[2] P. B. Blakieet al., Adv. Phys.57, 363 (2008).[3] N. N. Bogoliubov, J. Phys. (Moscow)11, 23 (1947).[4] E. Witkowska, M. Gajda, and K. Rzążewski, Phys. Rev. A79,

033631 (2009).[5] E. Witkowska, M. Gajda, and K. Rzążewski, Opt. Commun.

283, 671 (2010).[6] N. Metropoliset al., J. Chem. Phys.21, 1087 (1953).

[7] E. H. Lieb, Phys. Rev.130, 1616 (1963).[8] M. Brewczyket al., J. Phys. B37, 2725 (2004).[9] Ł. Zawitkowskiet al., Phys. Rev. A70, 033614 (2004).

[10] V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz.64, 1627(1973) [Sov. Phys. JETP37, 823 (1973)].

[11] T. Karpiuket al., Phys. Rev. A81, 013629 (2010).[12] J. Armijo et al., Phys. Rev. A83, 021605(R) (2011).

arXiv:1203.1811 (2012).[13] V. Guarreraet al., Phys. Rev. Lett.107, 160403 (2011).[14] S. Burgeret al., Phys. Rev. Lett.83, 5198 (1999).[15] J. Denschlaget al., Science287, 97 (2000).[16] C. Beckeret al., Nature Phys.4, 496 (2008).[17] C. Grosset al., Nature,464, 1165-1169 (2010).[18] J. J. Chang, P. Engels, and M. A. Hoefer, Phys. Rev. Lett.101,

170404 (2008).[19] M. Schmidtet al., arXiv:1203.3651 (2012).